Nonequilibrium phase transition in single-file transport at high crowding

Driven particle transport in crowded and confining environments is fundamental to diverse phenomena across physics, chemistry, and biology. A main objective in studying such systems is to identify novel emergent states and phases of collective dynamics. Here, we report on a nonequilibrium phase transition occurring in periodic structures at high particle densities. This transition separates a weak-current phase of thermally activated transport from a high-current phase of solitary wave propagation. It is reflected also in a change of universality classes characterizing correlations of particle current fluctuations. Our findings demonstrate that sudden changes to high current states can occur when increasing particle densities beyond critical values.

💡 Research Summary

The authors investigate a nonequilibrium phase transition in a one‑dimensional single‑file system of Brownian particles driven across a periodic potential. The model, termed the Brownian Asymmetric Simple Exclusion Process (BASEP), consists of hard‑sphere particles of diameter σ that experience an overdamped Langevin dynamics under a constant drag force f and a sinusoidal potential U(x)=U₀/2 cos(2πx/λ). Hard‑core exclusion enforces |x_i−x_j|≥σ, and periodic boundary conditions make the system homogeneous. The key observable is the instantaneous particle current density J(t)= (1/L)∑_i ẋ_i(t); its long‑time average J(ρ) is studied as a function of the particle density ρ=N/L.

In the zero‑noise limit (D = 0) the current‑density curve exhibits a sharp threshold at a critical density ρ_c. For ρ≤1 each potential well can host at most one particle; without thermal activation particles cannot surmount the barriers, so J=0. When ρ>1, multiple particles occupy the same well and form a “basic cluster” of size n_b(σ,f). This cluster is the largest mechanically stable aggregate that can sit in a potential minimum; its size follows from a minimal‑free‑space principle and is linked to Euler’s totient function. The number of wells required to accommodate such a cluster is ⌈n_b σ⌉, and the critical density is

 ρ_c = n_b / ⌈n_b σ⌉ .

Above ρ_c the system no longer remains jammed. Instead, clusters detach, translate across a barrier, and re‑attach in a periodic process that creates solitary wave packets (solitons). During one soliton period τ_sol the centre of mass of the whole system advances by exactly one wavelength λ, i.e. a total displacement Δ=λ. The number of solitons present is N_sol = (ρ−ρ_c) ⌈n_b σ⌉, so the current obeys a simple linear law

 J = (ρ−ρ_c) v_sol , v_sol = ⌈n_b σ⌉/τ_sol .

Thus, just above the transition the current grows linearly with density. For small σ where ⌈n_b σ⌉>1 a second characteristic density ρ* = ρ_c + 1/⌈n_b σ⌉² appears; when ρ exceeds ρ* the spacing between solitons matches the cluster size and the J‑ρ curve develops a pronounced kink, after which the slope increases sharply.

Finite thermal noise (D>0) smooths the transition. Thermal activation yields a small but non‑zero current for ρ<ρ_c, yet at high densities the soliton‑mediated transport dominates. The effect of noise is stronger for smaller particles; for σ=4/5 the transition remains visible up to D=0.05, whereas for σ=1/4 it disappears already at D≈0.05, and for σ=3/25 it is washed out already at D≈0.02.

Beyond the current‑density relation, the authors examine the universality class of current‑fluctuation correlations. In a comoving frame with velocity v(ρ)=J′(ρ), the current correlation function C(t)=⟨δj_cf(x,t) δj_cf(x,0)⟩ decays as a power law. If the curvature J″(ρ)=0 the system belongs to the Kardar‑Parisi‑Zhang (KPZ) class (C∼−t⁻⁴⁄³); if J″(ρ)≠0 it falls into the Edwards‑Wilkinson (EW) class (C∼−t⁻³⁄₂). Numerical simulations (σ=1/4, f=0.1, D=0.05) show that near ρ≈3.3 the second derivative is essentially zero, and the correlation decay follows the EW exponent, whereas at ρ≈3.05 where J″≈1.4 the decay follows the KPZ exponent. Hence the phase transition is accompanied by a change of universality class, reflecting that soliton transport behaves like unbiased diffusion of quasiparticles.

Phase diagrams are constructed in the σ‑ρ plane (for fixed f) and the f‑ρ plane (for fixed σ). The jammed (red) and current‑carrying (green) regions are separated by the line ρ=ρ_c(σ,f). Within the green region, a dark‑green sub‑area indicates linear J(ρ) and a bright‑green sub‑area indicates the nonlinear regime beyond ρ*. The basic cluster size n_b varies irregularly with σ for 0.35≲σ≲0.8, producing dips in ρ_c, while for larger drag forces f the dependence becomes monotonic and ⌈n_b σ⌉=1, leading to stepwise decreases of ρ_c with increasing f.

In summary, the paper demonstrates that a homogeneous single‑file system can undergo a genuine nonequilibrium phase transition from a weak‑current, thermally activated regime to a high‑current, soliton‑mediated regime without any external reservoirs or spatial inhomogeneities. The transition is governed by the maximal mechanically stable cluster that can be accommodated in the periodic energy landscape, and its location in parameter space is dictated by particle size and driving force. Above the threshold, solitons dominate transport, yielding an almost linear current‑density relation and an Edwards‑Wilkinson scaling of current fluctuations. The work provides a quantitative framework for predicting and controlling collective transport in crowded nano‑channels, ion channels, and other confined driven systems.